Extremal results for directed tree connectivity

TL;DR

This paper investigates the extremal properties of directed graphs with respect to generalized tree connectivity, focusing on minimally strongly connected digraphs and characterizing their sizes for various parameters.

Contribution

It introduces the concept of minimally generalized $(k, \, ext{ell})$-vertex and arc-strongly connected digraphs, providing size bounds and characterizations.

Findings

Determined minimum and maximum sizes of such digraphs.

Provided characterizations for specific pairs of parameters.

Extended classical connectivity concepts to generalized directed tree connectivity.

Abstract

For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $\kappa_{S,r}(D)$ and $\lambda_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$\kappa_k(D)= \min \{\kappa_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ Similarly, the generalized $k$-arc-strong connectivity of $D$ is defined as $$\lambda_k(D)= \min \{\lambda_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized $k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs. A digraph $D=(V(D), A(D))$ is called minimally generalized $(k, \ell)$-vertex (respectively, arc)-strongly connected if $\kappa_k(D)\geq \ell$ (respectively, $\lambda_k(D)\geq \ell$) but for any arc $e\in A(D)$, $\kappa_k(D-e)\leq \ell-1$ (respectively, $\lambda_k(D-e)\leq \ell-1$). In this paper, we study the minimally generalized $(k, \ell)$-vertex (respectively, arc)-strongly connected digraphs. We compute the minimum and maximum sizes of these digraphs, and give characterizations of such digraphs for some pairs of $k$ and $\ell$.